Directional Differentiability of a Continual Maximum Function of Quasidifferentiable Functions

NOT FOR QUOTATION WITHOUT PERMISSION OF THE _AUTHOR

DIRECTIONAL DIFFERENTIABILITY OF A CONTINUAL MAXIMUM FUNCTION OF QUASIDIFFERENTIABLE FUNCTIONS V . F . Demyanov

I.S. Z a b r o d i n J u n e 1 9 8 3 WP-83-58

V o r k i n q P z p e r e

a r e

i n t e r i m r e p o r t s on work o f the I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s and have r e c e i v e d o n l y l i m i t e d r e v i e w . Views o r o p i n i o n s e x p r e s s e d h e r e i n do n o t n e c e s s a r i l y r e p r e - s e n t t h o s e o f the I n s t i t u t e o r o f i t s N a t i o n a l Member O r g a n i z a t i o n s .

INTERNATIONAL IXSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a

PREFACE

Much recent work in optimization theory has been con- cerned with the problems caused by nondifferentiability.

Some of these problems have now been at least partially overcome by the definition of a new class of nondifferen- tiable functions called quasidifferentiable functions, and the extension of classical differential calculus to deal wit11 this class of functions. This has led to increased theoretical research in the properties of quasidifferen- tiable functions and their behavior under different con- di tions .

In this paper, the problem of the directional differen- tiability of a maximum function over a continual set of

quasidifferentiable functions is discussed. It is shown that, in general, the operation of taking the "continual"

maximum (or minimum) leads to a function which is itself not necessarily quasidifferentiable.

Andrze

_j

Wierzbicki Chairman

System

&

Decision Sciences

DIRECTIONAL DIFFERENTIABILITY OF A CONTINUAL MAXIMUM FUNCTION OF QUASIDIFFERENTIABLE FUNCTIONS V. F . ^Demyanov

I.S. Zabrodin

1. INTRODUCTION

Optimization problems involving nondifferentiable functions are reccgnized to be of great theoretical and practical sig- nificance. There are many ways of approaching the problems caused by nondifferentiabi.lity, Some of which are now quite well developed. while others still require much further work. A

comprehensive bibliography of publications concerned with non- differentiable optimization has recently been compiled [1]--

major contributors in this field include J.P. Aubin, F.H. Clarke, Yu.M. Ermoliev, J.B. Hiriart-Urruty, A.Ya. Kruger, S.S. Kutateladze, C. ~emargchal, B.

S

. Morduchovich, E .A. Nurminski, B. N. P~her~ichniy, R.T. Rockafellar, and J. Warga.

The notion of subgradient has been generalized to nonconvex functions in a number of different ways. One of these involves the definition of a new class of nondifferentiable functions

(quasidifferentiable functions) which has been shown to represent a linear space closed with respect to all algebraic operations as well as to the taking of pointwise maximum and minimum

[ 2 , 3 ]

.

This has led to the development of quasidifferential calculus--

a generalization of classical differential calculus--which may

be used to solve many new optimization problems involving non-

differentiability

[ 4

1 .

This paper d e a l s

w i t h

t h e problem of t h e d i r e c t i o n a l d i f - f e r e n t i a b i l i t y of a

maximum

f u n c t i o n o v e r a c o n t i n u a l

s e t

of q u a s i d i f f e r e n t i a b l e f u n c t i o n s .

I t w i l l b e

shown t h a t i n g e n e r a l t h e o p e r a t i o n of t a k i n g t h e " c o n t i n u a l "

maximum (minimum) l e a d s

t o a f u n c t i o n which

i s

i t s e l f n o t n e c e s s a r i l y q u a s i d i f f e r e n t i a b l e .

2 . AUXILIARY RESULTS

L e t

us c o n s i d e r a mapping

G : En

-

²

^{Em , where}

²

^Em

d e n o t e s t h e s e t of a l l s u b s e t s of

,E

. ^{Fix x}

₀ ^{E En}

^and

9

E En

, ^{Ogm=l. Choose}

y E G

( x

)

and i n t r o d u c e t h e s e t

0 1

W e

s h a l l d e n o t e t h e c l o s u r e

o f y

( y ) by

F

( y ) ,

i.

e . ,

The set

F

( y )

is

c a l l e d t h e s e t of f i r s t - o r d e r f e a s i b l e d i r e c t i o n s a t

the

p o i n t y

E

G(x

)

i n t h e d i r e c t i o n g .

0

R e m a r k 1.

I n

the

c a s e where

G

does n o t depend on x , t h e s e t

F

( y )

= F (x

, g , y ) does n o t depend on x and g , ^and

i s

0 0

a

cone c a l l e d

the

cone of f e a s i b l e d i r e c t i o n s a t y .

A

mapping

G i s

s a i d t o a l l o w f i r s t - o r d e r approximation a t a p o i n t x i n

the

d i r e c t i o n g

E En

, YgI-1 , i f , f o r an

0

a r b i t r a r y convergent sequence { y k } such t h a t

t h e f o l l o w i n g r e p r e s e n t a t i o n h o l d s

:

where vk

E

F ( x , g , y )

₀ ^I

akVk -

^{0 I !? 5 ( x} ₀ ⁾

I n what f o l l o w s

i t i s

assumed t h a t t h e mapping

G is

continuous ( i n t h e Hausdorff m e t r i c ) a t a p o i n t x and

0

allows f i r s t - o r d e r approximation a t x i n any d i r e c t i o n

0

g E E n , ^Ilgll

^{= 1}

.

I t i s

a l s o assumed t h a t f o r every

t h e s e t G(x)

i s

c l o s e d and s e t s G(x) a r e j o i n t l y bounded on

S ( X )

, i . e . , t h e r e e x i s t s an open bounded

s e t B C Em

0

such t h a t

L e t

us c o n s i d e r t h e f u n c t i o n f ( x )

=

max $ ( x , y )

YfG

( x )

where f u n c t i o n

@

( z )

=

4 ( x , y ) i s continuous i n z

= [

x , y ] on S 6 ( x 0 )

x B

and d i f f e r e n t i a b l e on

Z

i n any d i r e c t i o n

0

rl =

[ g , q ]

E En+,

, i . e . , t h e r e e x i s t s a f i n i t e l i m i t

Here

Suppose t h a t t h e f o l l o w i n g c o n d i t i o n s h o l d :

C.-t;.dition

1.

I f

^qk

-

^q

^{t h e n}

C o n d i t i o n 2 . L e t yk E R ( x

+

a k g )

,

yk

-

^y

-

s i n c e

0

G a l l o w s f i r s t - o r d e r a p p r o x i m a t i o n a t x .

,

^{t h e n}

0

I t i s assumed t h a t the q k l s

are

bounded.

C o n d i t i o n 3. F u n c t i o n 4 i s L i p s c n i t z i a n i n some neigh- borhood of t h e s e t Z

.

0

Then t h e f o l l o w i n g r e s u l t h o l d s .

T h e o r e m I . T h e f u n c t i o n _f is d i f f e r e n t i a b l e at the point x in the d i r e c t i o n _g a n d

0

a f ( x

a @ ( x

, Y )

O = sup SUP ⁰

ag ~ E R ( X ) ^O q f r ( y ) a [ g , q l

Proof. _{L e t} us d e n o t e by A t h e r i g h t - h a n d s i d e of (1). F i x y E R ( x ) and q E _{y ( y )}

.

^{Then y}

⁺

a q E G ( x + a g ) f o r

0 0

s u f f i c i e n t l y s m a l l a > 0 and

a o

^{( X} , Y ) f ( X

+

a g )

a

^@ ( x +ag , y + a q ) = f ( x + a 0

0 0 0

Hence

1

a o

^{(X , y )}

l i m

h ( a )

l i m

- - -

^{[ f ( x}

⁺

^{a g )}

^-

^{f ( x}

^{1 1}

⁰

a-tO a+O a o o

a t MI

S i n c e y E R ( x ) and q E y ( y ) a r e a r b i t r a r y t h e n

0

ao

c x 0 , y )

-

^{liln 1 ( a ) 2 sup}

a-+O SUP

;.fRix ) qEy ( y ) a [ q , q l

0

L e t qk

-

^q

^,

^{qk E y ( y )}

^.

^{Then q E r ( y )}

^.

^{I t} f o l l o w s from C o n d i t i o n 1 t h a t

But

a 4 ( x

, Y )

a 4 ( x

, Y )

-

a[ gt4*1

^<

4 E y ^{s u p} ( Y ) a [ g t s ⁰

I

S i n c e

r

( y ) = c l Y _{( y )}

,

then from ( 3 )

a @ ( x ⁰ , y ) a $ ( x ⁰ t y )

a o ^{( x o}

^{/ y )}

SUP 4 s u p s u p

* ( Y ) a [ g , q I W ( Y ) a ~ g ~ q l

GY

( Y )

a [

^{g , q l}

From

a q ( x , Y )

a w

, Y )

suy:

A

= s u p ⁰

+ r ( y ) a [ g l q i +Y ( Y Y a [ g , q l

and f 011 ows that

NOW l e t u s c h o o s e s e q u e n c e s { y k } and { a k } s u c h t h a t

The c o n d i t i o n s imposed on t h e mapping G and t h e con- t i n u i t y of t h e f u n c t i o n e n s u r e t h a t the f u n c t i o n

f

i s

c o n t i n u o u s a t x o x e n c e , from t h e e q u a l i t y f ( x +'ikg) = o ( x t C l , g

,

^pki ,

0 0

- -

o n e c a n c o n c l u d e t h a t f ( x ) = ~ ( x , y )

,

e y E R ( x )

.

0 0 0

Since the mapping G allows first-order approximation at x , ^{then yk}

⁼

- ^y

+

akqk + o (ak) , ^{where. qk}

0

E

r(?) ,

akqk

^---40

. From Conditions 2 and 3 the qk' s are bounded and

1.;:

function is Lipschitzian around Z . Without loss of

0

generality one can assume that qk - ^{q .} It is clear that q

E

r ( y ) . ^Hence

where

Since

@

is a Lipschitzian function, then

It follows from

(6)-(8)

that

.front wnicn it is clear that

a$(x ,Y)

T r n h(a)

SUP

SUP

o

= A .

Comparison of

( 5 )

and (9) now shows that lim h(a) exist-

a+c!

ana is equal to A , thus completing the proof.

Remark 2.

Equation (1) has been proved under some different

assumptions elsewhere

[ 5

1 (see also

[ 6

1 ,

$

10

)

. The case where

4

is differentiable was studied by Hogan

[ 7 ] .

3. QUASIDIFFERENTIMLE CASE

L e t us c o n s i d e r once a g a i n t h e f u n c t i o n f ( x = rnax Q ( x , y )

a yEG

( X I

where mapping G s a t i s f i e s t h e c o n d i t i o n s s p e c i f i e d e a r l i e r and f u n c t i o n Q ( 2 ) = Q ( x , y ) i s continuous i n z _on S ( x B

g ⁰

and q u a s i d i f f e r e n t i a b l e on 2. , i . e . , f o r a n y p a i n t z = [ x

0

-

0

o f Y o l

E ⁰

t h e r e e x i s t convex compacts

- ²⁰

^{( 2} _a ^{) C En+,} _and

_{a + ( ~}

_a

_c

_{E ~ + ~}

such t h a t

+

n i n [ (wltg) + ( w 2 f g )

I [ W ~ ' W ~ I ~ ~ ~ ( ~

⁰ )

I t

i s

a l s o assumed t h a t C o n d i t i o n s 2 and 3 a r e s a t i s f i e d . (Condition 1 f o l l o w s immediately from ( l l ) . ) Thus, a l l t h e c o n d i t i o n s of Theorem 1 a r e f u l f i l l e d and we a r r i v e a t

T h e o r e m 2. T h e f u n c t i o n f d e f i n e d b y ( 1 0 1 i s d i r e c t i o n a l l y d t j f s r e n t i a b l z a n d , m o r e o v e r ,

a f ( ~

¹

A = sup SUP

\

max [ ( v l I g ) + ( v 2 , q )

I

ag Y-(x o )

W ( Y ) 1

[ v l f v 2 1 e 2 Q ( x

-

₀ ^{, y )}

+

min ^[ (wS&) ⁺ (w2tq)

1 ^I / .

^{( 1 2 )}

[ w1,w21

e a 4

( x , y )

0

R e m a r k 3 . S i n c e y E R(x , y )

,

t h e f o l l o w i n g r e l a t i o n h o l d s :

0

SUP

\

^max ( v 2 ' q ) + n i n

I

( w 2 ' q ) j = 0 + r ( y )

1

~ ~ € 3 0 ^{( X} , y ) ^-

- Y ⁰ w2ES$y (x 0 I Y )

Here 39 (x ,y) and -

d $

(xo,y) are the projections of sets

- Y

0

- ^Y

ap (xo ,y) and a$ (xO ,y), respectively, onto Em . -

R e m a r k 4 .

Pshenichniy

18

1 considered the case where G (x) does not depend on x and F (x)

=

4 (x,y) is a directionally differentiable function for every fixed y Y , i.e., there exists

a9 (x,Y) 1

=

lim -

^[

^$(x+ag,y) -

^$

^(x,y)l

ag

a 4 0

a

Then

Under an additional assumption about the behavior of o(a,y) in (13) , it has been proved that

af (XI a4 (x,Y)

= max

ag yER (x) ag

It is clear that equation (14) differs from equation (12).

E x a m p l e I .

Let x E E l , ^{y E E 1} ,

G ( x ) G =

[-2,2] ,

4(x,y)

=

x - ^21y - ^xl , ^and

f(x)

=

max (x - 2 I y - X I ) -

yEl -2,21 It is clear that

Choose x

E

(-2,2) and verify equation (14) . We shall now compute the right-hand side of (14). Since

4(x,y)

=

x - ^{2 max {y} - ^{x , -y + x) ,} then for y

E R ( x ) =

{XI

it follows

[ 9

I ^that

a$ (x,Y)

=

g - 2 max E-g,g) .

a4

Hence f o r gl = +l

a +

( x , Y )

max

= 1 - 2 2 - 1 ,

F R ( x )

and f o r g 2 = -1

But from ( 1 6 ) i t

i s

c l e a r t h a t

Thus e q u a t i o n ( 1 4 ) d o e s n o t h o l d f o r any d i r e c t i o n g ( i n E l t h e r e a r e o n l y two d i r e c t i o n s g s u c h t h a t llgll = 1 : g = +1 and g = -1).

sow l e t u s v e r i f y e q u a t i o n ( 1 2 ) . Denote by D the t i g h t - hand s i d e of ( 1 2 )

.

The f u n c t i o n $ ( x , y ) i s q u a s i d i f f e r e n t i a b l e . From q u a s i d i f f e r e n t i a l c a l c u l u s [ 2 - 4 1 it f o l l o w s t h a t i f y =

x

t h e n o n e c a n c h o o s e a $ ( x , y )

-

= { ( 1 , O )

,

5 $ ( x , y ) = c o { ( - 2 ~ 2 ) ^I ( 2 1 - 2 )

1 -

For t h e f u n c t i o n f d e s c r i b e d by ( 1 5 ) w e have

Computing D :

I

D

= sup

1

( 1 . g )

+

( 0 . q )

+

inin [(wl.g) + (w2.q)1

j

-1 ^["l' w 2 lEcoI(-2,2) , ( 2 , 2 )

I

= g

+

^sup min [ ( ~ ~ - 9 ) + (w,

41.

^{( 1 8 )}

$El [ w1

,",I - s[

( - 2 , 2 )

,

( 2

, I

I t i s c l e a r from F i g u r e 1 t h a t f o r any g t h e second

term

on t h e r i g h t - h a n d s i d e of ( 1 8 ) i s e q u a l t o z e r o , i . e . , D = g

.

(The supremum i n ( 1 8 ) is a t t a i n e d a t q = g . )

F i g u r e 1.

T h u s , from ( 1 7 )

,

e q u a t i o n ( 1 2 ) is c o r r e c t ir, t h i s c a s e .

Remark 5 . When s o l v i n g p r a c t i c a l problems i n which i t i s r e q u i r e d t o minimize a max f u n c t i o n o v e r a c o n t i n u a l s e t of p o i n t s , t h i s maximum f u n c t i o n i s o f t e n d i s c r e t i z e d ( t h e con- t i n u a l s e t r e p l a c e d by a g r i d of p o i n t s ) . I n many c a s e s t h i s o p e r a t i o n i s a l e g i t i m a t e one

[ l o ] ,

b u t w e s h a l l show t h a t i n the c a s e where i s a q u a s i d i f f e r e n t i a b l e f u n c t i o n t h i s r e p l a c e m e n t may b e d a n g e r o u s .

L e t f a g a i n b e d e s c r i b e d by ( 1 5 )

.

D e f i n e f N a s

where

aN

= {x1 ^{I . =}

+ I

^I _Xk E [ - 2 , 2 1

.

T h i s f u n c t i o n h a s N l o c a l minima (see F i g u r e 2 1 , a l t h o u g h t h e o r i g i n a l f = x h a s no l o c a l minimum which i s n o t a l s o g l o b a l o n [ - 2 , 2 ]

.

T h i s d e m o n s t r a t e s t h a t t h e d i s c r e t i z a t i o n o f t h e max-type f u n c t i o n must b e c a r r i e d o u t v e r y c a u t i o u s l y .

F i g u r e 2

Ezample 2 . T h i s example i l l u s t r a t e s t h a t C o n d i t i o n 2 i s e s s e n t i a l t o o u r argument.

9 ( x , Y ) = x

-

² n i n i / ( x -

t 3 1 2

+ ( y

-

^{t ) 2}

tEE - 2 , 2 1

f ( x ) =

max

@ ( x , Y ) yEI - 2 , 2 1

I t i s c l e a r t h a t f o r x E ( - 2 , 2 )

,

I f y =

\3J;- ^,

t h e n t h e minimum i n ( 1 9 ) i s a c h i e v e d a t t = x

.

Take x = 0

.

^{Then R ( 0 ) = {O}}

^,

^{I'(y) = E l}

.

C o n s t r u c t a

0

q u a s i d i f f e r e n t i a l of t h e f u n c t i o n $ a t t h e p o i n t ( 0 , O )

.

By t h e r u l e s o f q u a s i d i f f e r e n t i a l c a l c u l u s one c a n choose

L e t u s d e n o t e by D t h e r i g h t - h a n d s i d e of e q u a t i o n ( 1 2 ) and e v a l u a t e it.

D = 3 ( g . ) = s u p min (wl.g + W2.q)

I

+El Lwltw21~co [ ( - 2 t O ) t ( 2 , O ) I

I

If g1 = 1 t h e n D(gl) = -1 if g 2 = -1 t h e n D ( g 2 ) = - 3

.

But i t i s c l e a r from ( 2 0 ) t h a t a f (O)/ag = g

.

Thus e q u a t i o n ( 1 2 ) d o e s n o t h o l d , and t h e r e a s o n i s t h a t C o n d i t i o n 2 i s n o t s a t i s f i e d . I n d e e d , t a k i n g an a r b i t r a r y s e q u e n c e xk = x

+

akg

0

where ak- + O

,

p u t t i n g , f o r example, g = 1

,

x = 0

,

w e

0

o b t a i n yk =

+

a v

k k t Y k E R ( x k )

.

For y

-

= 0 t h i s l e a d s t o R ( x k ) = {

V

^{a k }}

.

^But

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,

pp. 14-17, 1980.

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,

1 0 ( 2 7 )

,

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